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Properties

Label 2-450-45.34-c1-0-0
Degree $2$
Conductor $450$
Sign $-0.890 + 0.455i$
Analytic cond. $3.59326$
Root an. cond. $1.89559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (−1.41 + i)3-s + (0.499 + 0.866i)4-s + (−1.72 + 0.158i)6-s + (−3.85 − 2.22i)7-s + 0.999i·8-s + (1.00 − 2.82i)9-s + (−2.44 + 4.24i)11-s + (−1.57 − 0.724i)12-s + (−3.85 + 2.22i)13-s + (−2.22 − 3.85i)14-s + (−0.5 + 0.866i)16-s − 4.89i·17-s + (2.28 − 1.94i)18-s − 2.55·19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.816 + 0.577i)3-s + (0.249 + 0.433i)4-s + (−0.704 + 0.0648i)6-s + (−1.45 − 0.840i)7-s + 0.353i·8-s + (0.333 − 0.942i)9-s + (−0.738 + 1.27i)11-s + (−0.454 − 0.209i)12-s + (−1.06 + 0.617i)13-s + (−0.594 − 1.02i)14-s + (−0.125 + 0.216i)16-s − 1.18i·17-s + (0.537 − 0.459i)18-s − 0.585·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.890 + 0.455i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.890 + 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0631591 - 0.262130i\)
\(L(\frac12)\) \(\approx\) \(0.0631591 - 0.262130i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (1.41 - i)T \)
5 \( 1 \)
good7 \( 1 + (3.85 + 2.22i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.44 - 4.24i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.85 - 2.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.89iT - 17T^{2} \)
19 \( 1 + 2.55T + 19T^{2} \)
23 \( 1 + (2.12 - 1.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.22 + 2.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.224 - 0.389i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.34iT - 37T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.45 - 3.72i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.953 - 0.550i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.44iT - 53T^{2} \)
59 \( 1 + (0.275 + 0.476i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.4 - 7.17i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.34T + 71T^{2} \)
73 \( 1 - iT - 73T^{2} \)
79 \( 1 + (6.34 - 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.476 + 0.275i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (-9.35 - 5.39i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.86487514467247852870329375666, −10.55080715417199142841620009549, −9.940871033245227336638531962470, −9.273067970523295271616317925485, −7.33928340118937996512188685902, −7.00291445956419218020708527101, −5.94512310658161307055247502400, −4.77773364033562181998256893815, −4.13718781182922048881002259725, −2.74894366929963217297587744278, 0.13875971057856137959227205614, 2.33110614162747710902015764960, 3.36833338234159600190892725391, 4.99092170508301119295583134231, 5.98792794425868991490594197965, 6.30559143253877383345956130979, 7.66090198521380885658356075774, 8.771636458259107145395095275801, 10.11746789540829147053909842439, 10.59442798552812627520841757193

Graph of the $Z$-function along the critical line